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Construction of the topological toric theory

拓扑环面理论的构建

基本信息

批准号：
13640087
负责人：
MASUDA Mikiya
金额：
$ 2.11万
依托单位：
Osaka City University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

项目摘要

We developed the theory of toric varieties from the topological viewpoint. In these several years I worked with Professor Akio Hattori and found that geometrical properies of a torus manifold can be described in terms of a combinatorial object called a multi-fan. In particular, we found a neat formula describing the elliptic genus of a torus manifold in terms of the multi-fan associated with the torus manifold, and obtained a vanishing theorem saying that the level N elliptic genus of a torus manifold vanishes if the 1st Chern class of the manifold is divisible by N. As a corollary of this vanishing theorem, we obtained a result that if the 1st Chern class of a complete toric variety M of complex dimension n is divisible by N, then N must be less than or equal to n+1, and in case N=n+l, M is isomorphic to the complex protective space. This is a toric version of the famous Kobayashi-Ochiai or Mori's theorem.I invited Taras Panov from Moscow State University for a month and studied the equivariant cohomology of a torus manifold M and the cohomology of its orbit space. As a result, it turned out that when the cohomology ring of M is generated in degree two, the equivariant cohomology of M is a Stanley-Reisner ring and the orbit space of M has the same form as a convex polytope from a cohomological point of view. We also studied the case where M has vanishing odd degree cohomology. It turns out that this case is obtained by blowing down the previous case. Interestingly, the equivariant cohomology of M in this case provides a generalization of the Stanley-Reisner ring. The ring like this was already introduced by Stanley about ten years ago but we may think of our results as giving a geometrical meaning of the ring. Along this line, I proved a conjecture by Stanley about the h-vector of a Gorenstein* simplicial poset. The proof is purely algebraic but the idea stems from topology and this shows a close connection between combinatorics, commutative algebra and topology.

我们从拓扑的观点发展了复曲面簇的理论。在这几年里，我和服部昭夫教授一起工作，发现环面流形的几何性质可以用一个叫做多扇的组合对象来描述。特别地，我们发现了一个简洁的公式，用与环面流形相关联的多重扇来描述环面流形的椭圆亏格，并得到了一个消失定理，即如果环面流形的第一陈类可被N整除，则环面流形的N级椭圆亏格为零.作为这个消失定理的推论，我们得到了一个结果：如果复维数为n的完备复曲面簇M的第一陈类能被N整除，则N必小于或等于n+1，并且当N=n+1时，M同构于复保护空间.这是著名的Kobayashi-Ochiai定理或Mori定理的环面版本。我邀请了莫斯科州立大学的Taras Panov用了一个月的时间研究了环面流形M的等变上同调及其轨道空间的上同调。结果表明，当M的上同调环是二次生成的时，M的等变上同调环是Stanley-Reisner环，并且M的轨道空间从上同调的观点看具有与凸多胞形相同的形式.我们还研究了M的奇次上同调为零的情形。原来，这个案例是由前一个案例向下吹得到的。有趣的是，在这种情况下M的等变上同调提供了斯坦利-莱斯纳环的推广。像这样的环已经介绍了斯坦利约十年前，但我们可能会认为我们的结果，使几何意义的环。沿着这条线，我证明了斯坦利关于Gorenstein* 单纯偏序集的h-向量的一个猜想。证明是纯粹的代数，但想法源于拓扑结构，这表明了密切联系之间的组合，交换代数和拓扑结构。